Subquadratic Algorithms for Succinct Stable Matching

  • Daniel Moeller
  • Ramamohan Paturi
  • Stefan SchneiderEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)


We consider the stable matching problem when the preference lists are not given explicitly but are represented in a succinct way and ask whether the problem becomes computationally easier. We give subquadratic algorithms for finding a stable matching in special cases of two very natural succinct representations of the problem, the d-attribute and d-list models. We also give algorithms for verifying a stable matching in the same models. We further show that for \(d = \omega (\log n)\) both finding and verifying a stable matching in the d-attribute model requires quadratic time assuming the Strong Exponential Time Hypothesis. The d-attribute model is therefore as hard as the general case for large enough values of d.


Stable matching Attribute model Subquadratic algorithms Conditional lower bounds SETH 



We would like to thank Russell Impagliazzo, Vijay Vazirani, and the anonymous reviewers for helpful discussions and comments.


  1. 1.
    Abboud, A., Backurs, A., Williams, V.V.: Quadratic-time hardness of LCS and other sequence similarity measures. In: 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS). IEEE (2015)Google Scholar
  2. 2.
    Agarwal, P.K., Erickson, J., et al.: Geometric range searching and its relatives. Contemp. Math. 223, 1–56 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alman, J., Williams, R.: Probabilistic polynomials and hamming nearest neighbors. In: 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS). IEEE (2015)Google Scholar
  4. 4.
    Arkin, E.M., Bae, S.W., Efrat, A., Okamoto, K., Mitchell, J.S., Polishchuk, V.: Geometric stable roommates. Inf. Process. Lett. 109(4), 219–224 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Backurs, A., Indyk, P.: Edit distance cannot be computed in strongly subquadratic time (unless SETH is false). In: Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, 14–17 June 2015, pp. 51–58 (2015)Google Scholar
  6. 6.
    Bartholdi, J., Trick, M.A.: Stable matching with preferences derived from a psychological model. Oper. Res. Lett. 5(4), 165–169 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bhatnagar, N., Greenberg, S., Randall, D.: Sampling stable marriages: why spouse-swapping won’t work. In: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics, pp. 1223–1232 (2008)Google Scholar
  8. 8.
    Bogomolnaia, A., Laslier, J.F.: Euclidean preferences. J. Math. Econ. 43(2), 87–98 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bringmann, K.: Why walking the dog takes time: Fréchet distance has no strongly subquadratic algorithms unless seth fails. In: 2014 IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS), pp. 661–670. IEEE (2014)Google Scholar
  10. 10.
    Carmosino, M.L., Gao, J., Impagliazzo, R., Mihajlin, I., Paturi, R., Schneider, S.: Nondeterministic extensions of the strong exponential time hypothesis and consequences for non-reducibility. In: Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, pp. 261–270. ACM (2016)Google Scholar
  11. 11.
    Chebolu, P., Goldberg, L.A., Martin, R.: The complexity of approximately counting stable matchings. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX and RANDOM 2010. LNCS, vol. 6302, pp. 81–94. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Dabney, J., Dean, B.C.: Adaptive stable marriage algorithms. In: Proceedings of the 48th Annual Southeast Regional Conference, p. 35. ACM (2010)Google Scholar
  13. 13.
    Dobkin, D.P., Kirkpatrick, D.G.: A linear algorithm for determining the separation of convex polyhedra. J. Algorithms 6(3), 381–392 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69(1), 9–15 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gonczarowski, Y.A., Nisan, N., Ostrovsky, R., Rosenbaum, W.: A stable marriage requires communication. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1003–1017. SIAM (2015)Google Scholar
  16. 16.
    Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. Foundations of Computing Series. MIT Press, Cambridge (1989)zbMATHGoogle Scholar
  17. 17.
    Hershberger, J., Suri, S.: A pedestrian approach to ray shooting: shoot a ray, take a walk. J. Algorithms 18(3), 403–431 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Impagliazzo, R., Paturi, R., Schneider, S.: A satisfiability algorithm for sparse depth two threshold circuits. In: 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS), pp. 479–488. IEEE (2013)Google Scholar
  20. 20.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63, 512–530 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Knuth, D.E.: Stable Marriage and Its Relation to Other Combinatorial Problems: An Introduction to the Mathematical Analysis of Algorithms, vol. 10. American Mathematical Society, Providence (1997)Google Scholar
  22. 22.
    Matoušek, J.: Efficient partition trees. Discrete Comput. Geom. 8(1), 315–334 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Matoušek, J., Schwarzkopf, O.: Linear optimization queries. In: Proceedings of the Eighth Annual Symposium on Computational Geometry, pp. 16–25. ACM (1992)Google Scholar
  24. 24.
    Ng, C., Hirschberg, D.S.: Lower bounds for the stable marriage problem and its variants. SIAM J. Comput. 19(1), 71–77 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Patrascu, M., Williams, R.: On the possibility of faster SAT algorithms. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, 17–19 January 2010, pp. 1065–1075 (2010)Google Scholar
  26. 26.
    Razborov, A.A.: Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Math. Notes 41(4), 333–338 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Roth, A.E., Sotomayor, M.A.O.: Two-sided Matching: A Study in Game - Theoretic Modeling and Analysis. Econometric Society Monographs. Cambridge University, Cambridge (1990)CrossRefzbMATHGoogle Scholar
  28. 28.
    Segal, I.: The communication requirements of social choice rules and supporting budget sets. J. Econ. Theory 136(1), 341–378 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Smolensky, R.: Algebraic methods in the theory of lower bounds for boolean circuit complexity. In: Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, pp. 77–82. ACM (1987)Google Scholar
  30. 30.
    Williams, R.: A new algorithm for optimal constraint satisfaction and its implications. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1227–1237. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  31. 31.
    Williams, R.: Faster all-pairs shortest paths via circuit complexity. In: Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pp. 664–673. ACM (2014)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Daniel Moeller
    • 1
  • Ramamohan Paturi
    • 1
  • Stefan Schneider
    • 1
    Email author
  1. 1.University of CaliforniaSan Diego, La JollaUSA

Personalised recommendations